Unconditionally optimal convergence analysis of second-order BDF Galerkin finite element scheme for a hybrid MHD system
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Unconditionally optimal convergence analysis of second-order BDF Galerkin finite element scheme for a hybrid MHD system Yuan Li1 · Chunfang Zhai1 Received: 26 May 2020 / Accepted: 8 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, a second-order backward differentiation formula (BDF) scheme for a hybrid MHD system is considered. Being different with the steady and nonstationary MHD equations, the hybrid MHD system is coupled by the time-dependent Navier-Stokes equations and the steady Maxwell equations. By using the standard extrapolation technique for the nonlinear terms, the proposed BDF scheme is a semiimplicit scheme. Furthermore, this scheme is a decoupled scheme such that the magnetic field and the velocity can be solved independently at the same time as discrete level. A rigorous error analysis is done and we prove the unconditionally optimal second-order convergence rate O(h2 + (t)2 ) in L2 norm for approximations of the magnetic field and the velocity, where h and t are the grid mesh and the time step, respectively. Finally, the numerical results are displayed to illustrate the theoretical results. Keywords Magnetohydrodynamics equations · BDF scheme · Finite element method · Error analysis Mathematics Subject Classification (2010) 65M12 · 76W05
Communicated by: Jan Hesthaven Yuan Li
[email protected] 1
College of Mathematics and Physics, Wenzhou University, Wenzhou, 325035, People’s Republic of China
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Page 2 of 26
Adv Comput Math
(2020) 46:75
1 Introduction This paper focuses on a second-order backward differentiation formula (BDF) scheme for hybrid MHD equations which is governed by the following coupled system in Ω × (0, T ]: 1 ∂u − u + (u · ∇)u + ∇p + S b × curl b = f, ∂t Re div u = 0, 1 curl (curl b) − curl (u × b) = 0, Rm div b = 0,
(1.1) (1.2) (1.3) (1.4)
where T > 0 is some positive constant, Ω ⊂ is a bounded and convex domain, f is a given body force, three positive constants Re, Rm, and S denote the Reynolds number, magnetic Reynolds number, and the coupled number, respectively. In the above equations, the unknowns are the velocity u, the pressure p, and the magnetic field b. Unlike the steady or nonstationary MHD equations, the above system is coupled by the incompressible time-dependent Navier-Stokes equations and the steady Maxwell equations, which comes from the modelling of an industrial process when the magnetic phenomena are known to reach their steady state “infinitely” faster than the hydrodynamics phenomena [8]. To study the MHD system (1.1)–(1.4), the appropriate initial and boundary conditions are needed. In this paper, for simplicity, we consider: R3
u = 0,
b · n = q,
u(0) = u0 curl b×n=0
in Ω, on ∂Ω × [0, T ],
(1.5) (1.6)
where n denotes the unit outward normal vector on ∂Ω, and u0 satisfies div u0 = 0. The existence and uniqueness of the local strong solution to (1.1)–(1.6) were studied by Gerbeau and Le Bris in [6, 7]. As we know, it is difficult to find the analytical solutions to t
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